Interplanetary space probes
often make use of the "gravitational slingshot" effect to propel
them to high velocities. For example, Voyager 2 performed a close flyby of
Saturn on the 27th of August in 1981, which had the effect of slinging it
toward its flyby of Uranus on the 30th of January in 1986. Since gravity is a
conservative force, it may seem strange that an object can achieve a net gain
in speed due to a close encounter with a large gravitating mass. We might
imagine that the speed it gains while approaching the planet would be lost
when receding from the planet. However, this is not the case, as we can see
from simple consideration of the kinetic energy and momentum, which shows how
a planet can transfer kinetic energy to the spacecraft.

An extreme form of the
maneuver would be to approach a planet head-on at a speed v while the planet
is moving directly toward us at a speed U (both speeds defined relative to
the "fixed" Solar frame). If we aim just right we can loop around behind
the planet in an extremely eccentric hyperbolic orbit, making a virtual 180-degree
turn, as illustrated below.

The net effect is almost as if
we "bounced" off the front of the planet. From the planet's
perspective we approached at the speed U+v, and therefore we will also recede
at the speed U+v relative to the planet, but the planet is still moving at
(virtually) the speed U, so we will be moving at speed 2U+v. This is just
like a very small billiard ball bouncing off a very large one.

To be a little more precise,
conservation of kinetic energy and momentum before and after the interaction
requires

where subscripts 1 and 2
denote before and after, respectively. We eliminate U2 and solve
for v2 to give the result

Since m/M is virtually zero
(the probe has negligible mass compared with the planet), this reduces to our
previous estimate of v2 = v1 + 2U1.

Of course, most planetary
fly-bys are not simple head-on reversals, but the same principles apply for
any angle of interaction. Let's take the planet's direction of motion as the
x axis, and the perpendicular direction (in the orbital plane) as the y axis.
The probe is initially moving with a speed v relative to the solar reference
frame, in a direction approaching the oncoming planet at an angle theta. Two
views of this are shown below, one with respect to the planet's rest frame,
and the other with respect to the solar reference frame.

By drawing a simple
parallogram of speeds for the probe and planet intersecting at an arbitrary
angle q, and assuming we arrange for
a hyperbolic orbit symmetrical about the x axis (with respect to the planet's
rest frame), the probe's initial velocity vector with respect to the Sun's
rest frame is

and its final velocity vector
is

Thus its initial magnitude is
v1, and its final magnitude is

For example, suppose the
initial speeds of the probe and the planet happen to be exactly the same
(i.e., v1 = U). In this case the above relation reduces to

which confirms that when q = 0 we have v2 = 3v1,
which is our head-on reversal case. On the other hand, when q = p
we have v2 = v1, which stands to reason, because in
this case the probe and planet are going in the same direction at the same
speed. For a more realistic case, we can have the probe approach nearly perpendicular
to the planet's path (i.e., q = p/2) and swing just behind it. In that case
the probe gets deflected in the direction of the planet's travel, at an angle
given by the above formulas, and it's final speed is the square root of 5
(i.e., about 2.23) times its original speed.

If the planets were point
particles, then according to classical physics it would be theoretically
possible (in some rather contrived solar systems) for an object to acquire
infinite speed in finite time by looping repeatedly around a set of planets. Of
course, in practice the external gravitational field of a planet would
not be strong enough to "grab" the spaceship once it was traveling above
a certain speed. The limit is how fast you can loop around a planet without
dipping into its atmosphere too deeply (let alone crashing into it). Some
NASA missions have repeatedly skimmed the upper atmospheres of Venus and the
Earth in their maneuvers (cross- pollinating the environments?).

Conceivably, if we (or someone
else) ever found a star system consisting of multiple black holes orbiting
each other, it might be possible to apply this scheme to achieve relativistic
speeds, by looping around from one to the other. In this situation the achievable
speed limit would depend on how close a spaceship could pass without being
destroyed by tidal forces. Still, if the black holes were large enough, the
tidal forces even at the event horizon would be tolerable, although it
probably wouldn't be possible to have a controllable hyperbolic orbit pass
closer than, say 3m. Also, stopping the vehicle at the destination would be
difficult.